| L(s) = 1 | + 18·3-s − 38·5-s + 18·7-s + 243·9-s − 242·11-s − 66·13-s − 684·15-s − 920·17-s + 2.93e3·19-s + 324·21-s − 5.24e3·23-s + 2.65e3·25-s + 2.91e3·27-s − 1.26e4·29-s − 9.93e3·31-s − 4.35e3·33-s − 684·35-s + 5.99e3·37-s − 1.18e3·39-s + 2.42e4·41-s − 2.03e4·43-s − 9.23e3·45-s + 5.80e3·47-s − 3.30e4·49-s − 1.65e4·51-s + 4.07e4·53-s + 9.19e3·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.679·5-s + 0.138·7-s + 9-s − 0.603·11-s − 0.108·13-s − 0.784·15-s − 0.772·17-s + 1.86·19-s + 0.160·21-s − 2.06·23-s + 0.850·25-s + 0.769·27-s − 2.78·29-s − 1.85·31-s − 0.696·33-s − 0.0943·35-s + 0.720·37-s − 0.125·39-s + 2.25·41-s − 1.67·43-s − 0.679·45-s + 0.383·47-s − 1.96·49-s − 0.891·51-s + 1.99·53-s + 0.409·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 38 T - 1214 T^{2} + 38 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 18 T + 33382 T^{2} - 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 66 T - 135542 T^{2} + 66 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 920 T + 3050062 T^{2} + 920 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2932 T + 7100102 T^{2} - 2932 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5246 T + 19699918 T^{2} + 5246 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12600 T + 79348870 T^{2} + 12600 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9936 T + 53013118 T^{2} + 9936 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5996 T + 139663118 T^{2} - 5996 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 24244 T + 337184038 T^{2} - 24244 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 20360 T + 277332086 T^{2} + 20360 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5806 T + 419753950 T^{2} - 5806 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 40770 T + 1224329794 T^{2} - 40770 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18212 T + 455377462 T^{2} + 18212 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11398 T + 1131625826 T^{2} + 11398 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 65368 T + 3722340342 T^{2} + 65368 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 61446 T + 3799408318 T^{2} + 61446 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 53412 T + 4851340822 T^{2} - 53412 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 17122 T + 5872391094 T^{2} + 17122 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14304 T + 6955271542 T^{2} - 14304 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 58140 T + 11297620726 T^{2} + 58140 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 183056 T + 25458744990 T^{2} + 183056 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.703215572062441317245393192169, −9.404524401038371003844621421276, −8.965979035562144122186727757937, −8.508453198436435887854783588856, −7.86672593249534010871321628395, −7.71868700221813782379197477326, −7.28317316181811566040401622183, −7.04101858461517513295100667821, −5.96545176056967031047866079874, −5.76209606533280730462095125864, −5.08135487647118570739368664777, −4.48930055207176550383061119650, −3.82457059821943461645000536626, −3.68737250931299040823567454429, −2.96169957355391220021879034270, −2.40794943808158325867742960526, −1.78946640534122851517183591962, −1.27264775825290167090269437564, 0, 0,
1.27264775825290167090269437564, 1.78946640534122851517183591962, 2.40794943808158325867742960526, 2.96169957355391220021879034270, 3.68737250931299040823567454429, 3.82457059821943461645000536626, 4.48930055207176550383061119650, 5.08135487647118570739368664777, 5.76209606533280730462095125864, 5.96545176056967031047866079874, 7.04101858461517513295100667821, 7.28317316181811566040401622183, 7.71868700221813782379197477326, 7.86672593249534010871321628395, 8.508453198436435887854783588856, 8.965979035562144122186727757937, 9.404524401038371003844621421276, 9.703215572062441317245393192169
